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# Derivation of the Demand Curve in Terms of Utility Analysis

## Derivation of the Demand Curve in Terms of Utility Analysis

Dr. Alfred Marshall was of the view that the law of demand and so the demand curve can be derived with the help of utility analysis.
He explained the derivation of the law of demand

(i) In the case of a single commodity and
(ii) In the case of two or more than two commodities. In the utility analysis of demand, the following assumptions are made:

Assumptions
(i) Utility is cardinally measurable.
(ii) Utilities of different commodities are independent.
(iii) The marginal utility of money to the consumer remains constant.
(Iv) Utility gained from the successive units of a commodity diminishes.

(1)  Derivation of Demand Curve in the Case of a Single Commodity (Law  of Diminishing Marginal Utility)
Dr. Alfred Marshall derived the demand curve with the aid of law of diminishing marginal utility. The law of diminishing marginal utility states that as the consumer purchases more and more units of a commodity, he gets less and less utility from the successive units of the expenditure. At the same time, as the consumer purchases more and more units of one commodity, then lesser and lesser amount of money is left with him to buy other goods and services.
A rational consumer, before, while purchasing a commodity compares the price of the commodity which he has to pay with the utility of a commodity he receives from it. So long as the marginal utility of a commodity is higher than its price (MU x > P x ), the consumer would demand more and more units of it till its marginal utility is equal to its price MU x = P x or the equilibrium condition is established.
To put it differently, as the consumer consumes more and more units of a commodity, its marginal utility goes on diminishing. So it is only at a diminishing price at which the consumer would like to demand more and more units of a commodity.
Diagram/Curve

In fig. 2.4 (a) the MU x is negatively slopped. It shows that as the consumer acquires larger quantities of good x, its marginal utility diminishes. Consequently, at diminishing price, the quantity demanded of the good x increases as is shown in fig. 2.4 (b).
At X 1 , quantity the marginal utility of a good is MU 1 . This is equal to P 1 by definition. The consumer here demands OX 1 quantity of the commodity at P 1 price. In the same way X 2 quantity of the good is equal to P 2 . Here at P 2 price, the consumer will buy OX 2 quantity of commodity. At X 3 quantity the marginal utility is MU 3 , which is equal to P 3 . At P 3 , the consumer will buy OX 3 quantity and so on.
We conclude from above, that as the purchase of the units of commodity X are increased, its marginal utility diminishes. So at diminishing price, the quantity demanded of good X increases as is evident from fig. 2.4 (b). The rational supports the notion of down slopping demand curve that when price falls, other things remaining the same, the quantity demanded of a good increases and vice verse. (The negative section of the MU curve does not form part of the demand curve, since negative quantities do not make sense in economics).
(2) Derivation of the Demand Curve in the Case of Two or More than Two Commodities (Law of Equi-Marginal Utility):
The law of diminishing marginal utility can also be applied in case of two or more than two goods. When a consumer has to spend a certain given income on a number of goods, he attains maximum satisfaction when the marginal utilities of the goods are proportional to their prices as stated below.
MU x / P x = MU y / P y = ……….. MU n / P n
Derivation of Demand Curve:
In the fig. 2.5 (a), (b) and (c) given the money income, the price of X commodity (P x ) and the price of Y commodity (P y ) and constant marginal utility of money (MU m ), the demand curve derived is illustrated. The consumer allocates his money income between X and Y commodities to get OQ 1 units of good X and OY unit of good Y commodities because the combination correspondence to:
MU x / P x = MU y / P y = MU m
At the OM level (constant).
Diagram/Curve:

Let us assume that money income and price of Y commodity remain constant but the price of X commodity decreases. As a result of this money expenditure on commodity X rises resulting MU x / P x curve to shift towards the right. The consumer now allocates his income to OQ 2 quantity of X commodity and O y quantity of Y commodity because of the combinations correspondence to
MU x / P x = MU y / P y = MU m
(constant) at OM level.
Thus in response to decrease in the price from Px to Px 1 , the quantity demanded of a good X increases from OQ 1 to OQ 2 . The DD is a negatively sloped demand curve.
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